Method and system for formally analyzing the motion planning of a robotic arm based on conformal geometric algebra

ABSTRACT

Method and system for formally analyzing motion planning of a robotic arm based on conformal geometric algebra. The method includes determining specific structural and motion planning parameters of a robot, establishing a corresponding geometric model for the basic components and motion planning constraints of the robot based on a conformal geometric algebra theory, the established geometric model being described in a higher-order logic language, performing formal modeling for a motion process of the robot based on the established geometric model to obtain a logic model of the geometric relations involved in the motion process of the robot, obtaining a motion logic relationship corresponding to a constraint or attribute of a motion process to be verified of the robot, and verifying whether the motion logic relationship is correct. The method and system are used for analysis to improve the accuracy of the verification and reduce the complexity of the computations.

The present application claims priority to Chinese Patent ApplicationNo. 201610055938.2, filed with the State Intellectual Property Office ofthe People's Republic of China on Jan. 27, 2016 and entitled “Method andSystem for Formally Analyzing the Motion Planning of a Robotic Arm basedon Conformal Geometric Algebra,” which is included here in its entirety.

TECHNICAL FIELD

The present application relates to the technical field of computerscience, and specifically to a method and system for formally analyzingthe motion planning of a robotic arm based on conformal geometricalgebra.

BACKGROUND

Conformal Geometric Algebra (CGA), as an advanced geometricrepresentation and computing system, provides a concise, intuitive, andunified homogeneous algebraic framework for classical geometry. CGArepresents the reference origin (e₀) and a point at infinity (e_(∞)) byadding two dimensions, so that the Euclidean space is embedded into aconformal space, and gives it the structure of a Minkowski innerproduct, not only preserving the Grassmann structure of an outer productin homogeneous space, but also making the inner product possess adefinite geometric meaning characterizing the basic metric of thedistance, angle, and so on. CGA not only succeeds in solving the problemof how to accomplish geometric computations with geometric language, butalso plays a key role in solving geometric problems in manyhigh-technology fields, such as engineering and computer science. Inrobotics, CGA is different from the previously used algebra in that anobject involved in its computation is a geometry instead of a number.The object in robot research is a geometric relation formed by a systemestablished based on a basic geometry. Therefore, conformal geometricalgebra is unique in robot research.

At present, modeling, computation and analysis are performed using CGAas a mathematical tool. In this process, computer-based numericalcomputation and analysis and computer algebra systems (CASs) (e.g.,Maple, CLUCaic, Gaalop, etc.) are usually used. However, neither ofthese two methods can completely ensure the correctness and accuracy ofthe results. Because the number of iterations needed in the computationis limited by the computer memory and floating point numbers, numericalcomputation and analysis cannot completely ensure the accuracy of theresults. However, although the symbolic method provided by CASs canaccurately deduce the solution to a symbolic expression using a corealgorithm, the algorithm for computing an enormous set of symbols hasyet to be verified and there is a shortcoming in the processing ofboundary conditions. Thus, the results that are obtained may still beproblematic.

In recent decades, formal methods are widely used in many fields. Withthe development of basic research and the promotion of technologicalprogress, new methods and new tools are continuously emerging, and havegradually improved into mature and highly reliable verificationtechnologies. The main idea behind such technologies is to prove, basedon mathematical theory, that the system that has been designed meets thesystem specifications or has the desired properties. Compared with a penand paper-based manual analysis and the above traditional methods,formal methods can increase the chances of finding small but crucialerrors in early designs, according to the stringency of mathematicallogic.

The problems mentioned earlier arise when modeling, computation, andanalysis are performed using CGA as a mathematical tool. A formalanalysis of CGA theory is an ideal way to prevent such problems. BecauseCGA is unique in robot research, how to combine CGA with a formal way isan urgent technical problem that needs to be solved in robot research.

SUMMARY

Thus, the purpose of the present application is to provide a method andsystem for formally analyzing the motion planning of a robotic arm basedon conformal geometric algebra, in order to solve the problems ofcomplex computations and inaccurate results in the prior art whenanalyzing the motion planning of a robotic arm.

One aspect of the present application concerns the provision of a methodfor formally analyzing the motion planning of a robotic arm based onconformal geometric algebra, which includes:

determining the specific structural parameters and motion planningparameters of a robot;

establishing a corresponding geometric model for the basic componentsand motion planning constraints of the robot according to the specificstructural parameters and the motion planning parameters based on aconformal geometric algebra theory, wherein the established geometricmodel is described in a higher-order logic language; performing formalmodeling for a motion process of the robot based on the establishedgeometric model, to obtain a logical model of the geometric relationsinvolved in the motion process of the robot;

obtaining a logical relationship about motion corresponding to aconstraint or attribute of a motion process to be verified of the robot,using the above model;

verifying whether the logical relationship about motion is correct; ifit is correct, this indicates that the logical model of the geometricrelations meets the constraint of the motion process to be verified orhas the attribute of the motion process to be verified; otherwise, it isindicated that the logical model of the geometric relations does notmeet the constraint of the motion process to be verified or does nothave the attribute of the motion process to be verified.

Another aspect of the present application is the provision of a systemfor formally analyzing the motion planning of a robotic arm based onconformal geometric algebra, which includes:

a parameter determining module, configured for determining the specificstructural parameters and motion planning parameters of a robot;

a module for the establishing of robot basic geometric logical models,configured for establishing a corresponding geometric model for thebasic components and motion planning constraints of the robot accordingto the specific structural parameters and the motion planning parametersbased on a conformal geometric algebra theory, wherein the establishedgeometric model is described in a higher-order logic language;

a module for the establishing of a logical model of the geometricrelations in the motion process of the robot, configured for performingformal modeling for a motion process of the robot based on theestablished geometric model, to obtain a logical model of the geometricrelations in the motion process of the robot;

a module for the construction of logical propositions, configured forobtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relations;

a verification module, configured for verifying whether the motion logicrelationship is correct; if the motion logic relationship is correct,this indicates that the logical model of geometric relations meets theconstraint of the motion process to be verified or has the attribute ofthe motion process to be verified, or if the motion logic relationshipis not correct, this indicates that the logical model of geometricrelations does not meet the constraint of the motion process to beverified or does not have the attribute of the motion process to beverified.

Another aspect of the present application is the provision of a storagemedium. The storage medium is configured for storing executable programcode which, when executed, performs the above method for formallyanalyzing the motion planning of a robotic arm based on conformalgeometric algebra.

Another aspect of the present application is the provision of anapplication program. When executed, the program performs the abovemethod for formally analyzing the motion planning of a robotic arm basedon conformal geometric algebra.

Another aspect is the provision of an electronic device, which includes:

a processor, a memory, a communication interface, and a bus,

wherein the processor, the memory, and the communication interface areconnected and communicate with each other through the bus,

the memory stores executable program code,

the processor executes a program corresponding to the executable programcode by reading the executable program code stored in the memory, toperform the above method for formally analyzing the motion planning of arobotic arm based on conformal geometric algebra.

Unlike the traditional method, in the technical solutions provided inthe above embodiment, after a logical model of geometric relations isobtained by performing the modeling in a formal way, a motion logicrelationship corresponding to a constraint or attribute of a motionprocess to be verified is obtained according to the model, and whetherthe above motion logic relationship is correct is verified. The abovemethod is accurate and complete for the nature of the verification,since correctness is verified using mathematical methods. In addition,CGA can perform the modeling and processing on geometric elements suchas points, lines, planes, circles, and spheres, and the rotation andtranslation of these geometric elements in a unified manner. It offersgreat advantages in dealing with problems of robot kinematics and motionplanning, and can improve dimensions for solving problems, thussimplifying the coupling in robot computations, thereby reducing thecomplexity of the computations. In view of the above two aspects, whenthe technical solutions provided by the above embodiments are applied toanalyze the motion planning of a robotic arm, the complexity of thecomputations can be reduced while the accuracy of the verification isimproved. Thus, the respective advantages of CGA and formal methods aregained at the same time, and the synergy of both is exploited.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to more clearly describe the embodiments of the presentapplication and the technical solutions in the prior art, drawings thatneed to be used in embodiments and the prior art will be brieflydescribed below. Obviously, the drawings provided below are for onlysome embodiments of the present application; those skilled in the artcan also obtain other drawings based on these drawings without anycreative efforts.

FIG. 1 is a flowchart of the formal modeling of the grasping of anobject by a robot, based on CGA in the present application;

FIG. 2 is a flowchart of the verification of an algorithm of thegrasping of an object by a robot in the present application;

FIG. 3 is a schematic diagram of the computing of a target circle Z_(t)in the present application;

FIG. 4 is a schematic diagram of the computing of a robot gripper circleZ_(h) in the present application;

FIG. 5 is a schematic diagram of the solving of a rotation operator anda translation operator in the present application.

DETAILED DESCRIPTION

In order to make the objectives, technical solutions, and advantages ofthe present application clearer and more understandable, the presentapplication will be described in more detail below with reference to theappended drawings and embodiments. Obviously, the described embodimentsare only some, and not all, of the embodiments of the presentapplication. All other embodiments obtained based on the embodiments ofthe present application by those skilled in the art without any creativeefforts fall into the scope of protection defined by the presentapplication.

Various embodiments of the present application will be described belowin more detail with reference to the drawings. In the drawings, the sameelements are denoted by identical or similar reference numbers. For thesake of clarity, various parts of the drawings are not drawn to scale.

The currently developed conformal geometric algebra system cantheoretically be extended to any dimensional space, and is mainlyapplied to the 5-dimensional conformal space. The conformal geometricalgebra provides a Grassmann structure representing a geometry, aunified effect representing the geometric transformation, and a bracketsystem and an invariant system representing geometric invariants. Itdemonstrates significant advantages in the aspects of geometric dataprocessing and geometric computing. Under the framework of geometricalgebra, different transformation representation and computation formshave uniformity. After a geometric space transformation, the geometricmeanings and geometric characteristics of the original geometric objectcan still be maintained. The object of robot research is a geometricrelation formed by a system established based on a basic geometry.Different from the previous algebra, the computational object ofconformal geometric algebra is a geometry rather than a number.Therefore, conformal geometric algebra is unique in robot research.Conformal geometric algebra can also be used to collectively constructtraditional methods of robot research, such as Lie Group and Liealgebra, the Wu method, Clifford algebra, and Spin vectors, to beapplied to the analysis and synthesis of the mechanism, which promotesthe development of mechanisms, forms a universal theoretical method, andreduces the complexity of modeling and deduction computing inmechanisms. Various geometric elements in a space can be expressed asspherical coordinates by extending the dimension, and a relativelysimple mathematical model can be obtained in the modeling of a complexmulti-degree-of-freedom parallel mechanism.

Under the CGA framework, geometric representation and construction canbe performed based on an inner product and an outer product,respectively. The geometric representation based on an outer productmainly reflects a mutually constructive relationship between thegeometries of different levels, while the representation based on aninner product may be used to construct a corresponding parametricequation through metrics characterizing parameters such as a distance,an angle, and the like. For any k-order blade A, the parametric equationbased on an outer product is X∧A=0, and that based on an inner productis X·A=0. Representations of both can be converted into each otherthrough a dual operation between the inner and outer products. Table 1gives the geometric representations established based on an innerproduct and an outer product. The two representations are respectivelyreferred to in many studies as a standard representation and a directrepresentation.

TABLE 1 CGA basic geometric representation standard direct geometryrepresentation representation point$P = {x + {\frac{1}{2}x^{2}e_{\infty}} + e_{0}}$ sphere$S = {P - {\frac{1}{2}r^{2}e_{\infty}}}$ S* = P₁ ∧ P₂ ∧ P₃ ∧ P₄ plane π= n + de_(∞) π* = P₁ ∧ P₂ ∧ P₃ ∧ e_(∞) circle Z = S₁ ∧ S₂ Z* = P₁ ∧ P₂ ∧P₃ line L = π₁ ∧ π₂ L* = P₁ ∧ P₂ ∧ e_(∞) point pair P_(P) = S₁ ∧ S₂ ∧ S₃P_(P)* = P₁ ∧ P₂

In table 1, x, n are marked in hold, and represent vectors in Euclideanthree-dimensional space:x=x ₁ e ₁ +x ₂ e ₂ +x ₃ e ₃

In addition, x²=x₁ ²+x₂ ²+x₃ ² in Table 1 is a standard inner product inthe Euclidean space, that is, the scalar product, which can be describedby the scalar product function in a multidimensional vector library. Thee₁, e₂, and e₃ are the unit orthogonal bases in Euclideanthree-dimensional space. In the direct representation method, ahigh-dimensional geometry is established mainly by connecting points{Pi} on the geometry with an outer product “∧”. For example, a spherecan be represented by four points on the sphere. The meaning of theouter product in the standard representation method is different, andindicates an intersection between geometries. For example, a circle canbe formed by an intersection of two spheres.

Specifically, an embodiment of the present application provides a methodfor formally analyzing the motion planning of a robotic arm based onconformal geometric algebra. In this method, the specific structuralparameters and motion planning parameters of a robot are firstdetermined. Then, a corresponding geometric model is established for thebasic components and motion planning constraints of the robot accordingto the above specific structural parameters and motion planningparameters, based on conformal geometric algebra theory, wherein theestablished geometric model is described in a higher-order logiclanguage. Formal modeling is performed for the motion process of therobot based on the established geometric model, to obtain a logicalmodel of the geometric relations involved in the motion process of therobot. Then, a motion logic relationship corresponding to a constraintor attribute of a motion process to be verified of the robot isobtained, using the model. Finally, whether the above motion logicrelationship is correct or not is verified. If the motion logicrelationship is correct, this indicates that the above logical model ofgeometric relations meets the constraint of the motion process to beverified or has the attribute of the motion process to be verified. Ifthe motion logic relationship is not correct, this indicates that theabove logical model of geometric relations does not meet the constraintof the motion process to be verified or does not have the attribute ofthe motion process to be verified.

It is worth mentioning that the constraints or attributes of each motionprocess of the robot to some degree reflect the control and theoperations that can be realized, and the control and the operations thatcannot be realized of the robot during this motion process, and thelike. To this end, it can be considered that the constraints orattributes of each motion process of the robot correspond to a kind ofmotion logic relationship. In addition, different robots have differentgeometric features, and different geometric features have an impact onthe motion logic relationship. Therefore, the logical model of geometricrelations for the robot needs to be used when the motion logicrelationship corresponding to the constraint or attribute of each motionprocess of the robot is determined. For example, two gripper ends of arobot gripper can be originally opened by 180 degrees, but due to thelimitation of the work space, the two gripper ends can only be opened by120 degrees at a certain position in the space such as a position Q;otherwise, the robot gripper will collide with other objects in the workspace. At this time, 120 degrees is a constraint of the robot gripper atthe position Q; that is, when the robot gripper moves in the work space,if it moves to the position Q, the degree of the opening between the twogripper ends must be adjusted from 180 degrees to 120 degrees, otherwisea collision event will occur.

Furthermore, from a mathematical point of view, the motion logicrelationship corresponding to the constraints or attributes of themotion process to be verified of the robot, which was obtained using thelogical model of geometric relations, can be understood as a logicalproposition composed of the above model, and the constraints orattributes of the motion process to be verified.

Specifically, a logic reasoning engine can be used to verify whether theabove motion logic relationship is correct.

In an implementation of the present application, a higher order logictheorem prover (HOL-Light) can be introduced in the process from formalmodeling for the motion process of the robot based on the establishedgeometric model so as to obtain a logical model of geometricrelationships involved in the motion process of the robot, to verifyingwhether the above motion logic relationship is correct by using theabove logic reasoning engine. Specifically, a CGA formal model isestablished using HOL-Light, and formal modeling is performed for themotion process based on the established CGA formal model, to obtain alogical model of geometric relations. Further, formal modeling isperformed for constraints or attributes of the motion process to beverified to obtain the above motion logic relationship. Finally, whetherthe formal model of the above motion logic relationship is correct isverified based on the logic reasoning engine in the CGA formal model.

FIG. 1 is a flow chart of the formal modeling and verification of thegrasping of an object by a robot using a CGA-based method proposed inthe present application. The main idea in the operating of a robot usingthis method is the geometric motion transformation representation ofCGA. However, the transformed geometric object in the presentapplication is a reference circle rather than a reference point. Thegeometric features of a robot gripper and an object that is grasped areindicated by circles. In the present application, the high-order logictheorem prover HOL-Light is used as a formal tool. HOL-Light is one ofthe most popular theorem provers. Not only is there a large researchteam and user group on HOL-Light, but also a substantial body ofmathematical libraries, such as a real number analysis library, atranscendental function library, an integral-differential library, and aseries of efficient proof strategies.

Specifically, the specific structural parameters and the motion planningparameters of robotic components are first determined. The specificstructural parameters could be joint parameters, connecting rodparameters, a reachable range, a plane on which a joint is located, andso on. The motion planning parameters could be translation vectors,rotation angles, and others. Then, a corresponding geometric model isestablished for the robotic components and the motion planningconstraints, based on the conformal geometric algebra theory. Forexample, a point model can be established based on joint information, aline model can be established based on the connecting rod parameters, asphere model can be established based on the reachable range, a planemodel can be established based on the plane on which a joint is located,a rigid body motion model can be established based on the motionplanning parameters, and so on. In the grasping of an object by serialrobotic arms as the motion process to be verified, constraints,attributes, and other information describing this motion process aredetermined to be inputs to HOL-Light to verify this motion process. Forexample, whether the robotic arm securely grasps the object that isgrasped is determined through the attribute information of the motionprocess to be verified, so as to achieve the target verification.

It should be noted that the motion process of the robot includes notonly the motion process of the grasping of an object by the robot, butalso other motion processes, such as a movement process of the robot'sarm and the like. The present application only takes the process of thegrasping of an object as an example for illustration, which does notlimit the present invention.

As shown in FIG. 1, the present application proposes a method forformally analyzing the motion planning of a robotic arm based onconformal geometric algebra, which includes:

establishing a corresponding geometric model for the basic componentsand motion planning constraints of a robot based on a conformalgeometric algebra theory, describing the established geometric modelusing a higher-order logic language, and forming a basic geometric logicmodel system for the robot;

determining the specific structural parameters and motion planningparameters of a robot;

performing formal modeling for a specific motion process of the robotbased on the basic geometric logic model system, to obtain a logicalmodel of the geometric relations involved in the motion process of therobot;

describing a constraint or attribute of a motion process of the robot tobe verified using a logic formula;

forming a logic proposition using the geometric relationship logic modelof the specific motion process of the robot and the constraint orattribute of the motion process to be verified;

verifying whether the logic proposition is correct using a logicreasoning engine; if the logic proposition is correct, this indicatesthat the model meets the constraint or has the attribute, or if thelogic proposition is not correct, this indicates that the model does notmeet the constraint or does not have the attribute.

It can be understood by those skilled in the art that the basiccomponents of the robot are generally described by the specificstructural parameters of those components, and that the motion planningconstraints of the robot are generally described by the motion planningparameters of the motion planning. For this purpose, the above specificstructural parameters and motion planning parameters need to bedetermined first before a geometric model corresponding to the basiccomponents and the motion planning constraints of the robot isestablished, such that the geometric model corresponding to the basiccomponents and the motion planning constraints can be established moreclearly and effectively.

In view of the above, generally the specific structural parameters andmotion planning parameters of the robot have been determined when thecorresponding geometric model is established.

Conformal geometric algebra theory is the theoretical basis of a methodfor formally analyzing the motion planning of a robot. The method forformally analyzing the motion planning of a robot may realize thereasoning and calculation of a logic representation and the proving of aformal system for a conformal geometric algebra and method for formallyanalyzing the motion planning of a robot using the logic reasoningengine, for a general method that can be used to analyze specificproblems concerning the robot.

Specifically, after the corresponding geometric model for the basiccomponents and motion planning constraints of a robot is establishedbased on the conformal geometric algebra theory, a plurality ofgeometric models could be obtained, and there would be an associationbetween these geometric models. Thus, we can understand that thesegeometric models can form the basic geometric logic model system of arobot. Therefore, the establishment of the logical model of geometricrelations in the motion process can subsequently be understood as amodeling processing based on the above basic geometric logical modelsystem.

Those skilled in the art can understand that, from a mathematical pointof view, the constraints or attributes of the motion process of therobot can be described through a logical formula, so that the logicalrelationship contained in the motion process of the robot can bepresented more intuitively than before. This approach is also morebeneficial than other approaches to obtain the motion logic relationshipcorresponding to the constraints or attributes of the motion process.

In view of this, in an implementation of the present application, when amotion logic relationship corresponding to the constraints or attributesof the motion process to be verified of the robot is obtained using thelogical model of geometric relations, a motion logic relationshipcorresponding to the constraints or attributes of the motion process tobe verified can be obtained using the above logical model of geometricrelations. This can be done on the basis of the logical formulacorresponding to the constraints or attributes of the motion process tobe verified. Alternatively, a logical proposition formed by the abovelogical model of geometric relations and the constraints or attributesof the motion process to be verified can be obtained by referring to thelogical formula corresponding to the constraints or attributes of themotion process to be verified.

Specifically, performing formal modeling for a motion planning processrelating to the planning for a specific motion of a robot based on thebasic geometric logic model system, includes performing formal modelingfor a motion planning process involving the grasping of an object by therobot.

Preferably, in the above process of “determining the specific structuralparameters and motion planning parameters of a robot,” the specificrobot is an n-degree-of-freedom serial robot arm with a T-shaped robotgripper at the end of it.

Preferably, in the above process of “performing formal modeling for aspecific motion process of the robot based on the basic geometric logicmodel system, to obtain a logical model of geometric relations in thespecific motion process of the robot”, the specific motion planning ofthe robot is a motion planning process involving the grasping of anobject by the robot arm.

Preferably, in the above process of “describing a constraint orattribute of the motion process of the robot to be verified using alogical formula,” the attribute of the motion process to be verified ofthe robot is that the robot gripper successfully and firmly grasps theobject.

The established geometric model includes a point model abstracted from arobot joint, a sphere model abstracted from a reachable range of an endof a robot joint, a plane model abstracted from an auxiliary plane onwhich a robot joint is located, and a line model formed by connectingrobot joint points, a geometric intersection model abstracted from theconstraint relationship of the basic components of a robot, a geometricpure rotation model abstracted from the rotation motion of the basiccomponents of a robot, a geometric pure translation model abstractedfrom the translation motion of the basic components of a robot, ageometric rigid body motion model abstracted from the reaching of adesired position by an end of a robot joint, a distance model betweengeometries abstracted from the metric relationship of the basiccomponents of a robot, and a geometric angle model abstracted from anangle of the basic components of a robot.

The above models are described in detail below. The geometry includes apoint, a line, a plane, a circle, a sphere, and a point pair, which areused to represent a basic component of a robot.

The following is the point model abstracted from a robot joint:S=s+½s²e_(∞)+e₀

The above model can be formalized in HOL-Light, specifically as follows:|−•s.point_CGAs=s$1% mbasis{1}+s$2% mbasis{2}+s$3% mbasis{3}+(&1/&2*(sdot s)) % null_inf+null₊zero

In this model, the robot joint is abstracted into a point, and the abovemathematical expression is a standard expression of a point in conformalgeometric algebra. The function point_CGA in this model represents themapping relationship of a point from the three-dimensional Euclideanspace to the five-dimensional conformal space.

In the above model expression, s=s₁e₁+s₂e₂+s₃e₃ represents a point inthe three-dimensional Euclidean space, e₁, e₂ and e₃ are unitorthonormal bases in the three-dimensional Euclidean space, s₁, s₂, ands₃ are coefficients, and S is an expression of the point s mapped fromthe Euclidean space into the conformal space. The input variable of thefunction point_CGA is a three-dimensional vector in the Euclidean space,and its return value is a multivector in the five-dimensional conformalspace. Since the conformal geometric algebra can be constructed from thegeometric algebra C1_(4,1), its data type is defined as areal{circumflex over ( )}(4,1)multivector. s$n represents the nthelement of the multidimensional vector s, the mbasis function representsthe basic blade function, the dot function represents the scalar productin the Euclidean space, and the null_inf and null_zero functionsrespectively represent zero vectors e₀ and e_(∞):

${e_{0} = {\frac{1}{2}\left( {e_{-} - e_{+}} \right)}},{e_{\infty} = {e_{-} + e_{+}}},$

wherein e₀ represents an origin point, e_(∞) represents a point atinfinity, and e₊, e⁻ can be replaced by e_(∞), e₀ to more compactlyrepresent a point in the conformal space. In the present application,base vectors e₊, e⁻ are defined as mbasis {4} and mbasis {5},respectively representing the fourth base vector and the fifth basevector in CGA space. The following is a formal definition:|−null_zero=(&1/&2)%(mbasis{5}−mbasis{4})∧null_inf=(mbasis{5})+(mbasis{4})

The function point_CGA in this model represents the mapping relationshipof a point from the three-dimensional Euclidean space to thefive-dimensional conformal space.

The following is a sphere model abstracted from a reachable range of anend of a robot joint:

$S = {P - {\frac{1}{2}r^{2}{e_{\infty}.}}}$

The above model can be formalized in HOL-Light, specifically as follows;|−∀p r. sphere_CGAp r=point_CGAp−((&1/&2)*(r pow 2))% null_inf

The reachable range of an end of a robot joint is abstracted into asphere S, whose center is the joint and whose radius is a connectingrod. The above mathematical expression is the standard expression of asphere in conformal geometric algebra. The input variables p, r of thefunction sphere_CGA respectively represent the center and the radius,wherein the center can be represented by a three-dimensional vector inthe Euclidean space.

The following is a plane model abstracted from an auxiliary plane onwhich a robot joint is located: π=n+de_(∞).

The above model can be formalized in HOL-Light, specifically as follows:|−∀n d. plane_CGAn d=n$1% mbasis{1}+n$2%mbasis{2}+n$3%mbasis{3}+d%null_inf

The auxiliary plane on which a robot joint is located is abstracted intothe plane π, and the above mathematical expression is a standardexpression of a plane in conformal geometric algebra. The inputvariables n and d of the function plane_CGA respectively represent thenormal vector of the auxiliary plane and the distance of the auxiliaryplane to the origin point. The normal vector of the auxiliary plane canbe represented by a three-dimensional vector in the Euclidean space.

The following line model is constructed by connecting the robot jointsof a robot:L°=A∧B∧e _(∞).

The above model can be formalized in HOL-Light, specifically as follows:|−∀A B.line_direct_CGAA B=A outer B outer null_inf

The above mathematical expression is an expression of a line L* formedby any two points in conformal geometric algebra, which can be used torepresent a line constructed by the joint points of a robot or someother auxiliary line, wherein A and B respectively represent two pointsin the conformal geometric space. The model is used to abstract theconnecting rod between two joints into a line. The above expression is adirect expression of a line in conformal geometric algebra. The inputvariables a and b of the function line_direct_CGA respectively representtwo points on the line and the data type is a real{circumflex over( )}(4,1)multivector, wherein the function outer represents an outerproduct operation in geometric algebra, and the outer product operationcan realize a construction from a low-dimensional geometry to ahigh-dimensional geometry, that is, a dimension-expanding operation.

The geometric intersection model o is abstracted from a constraintrelationship of the basic components of a robot: o=o₁∧o₂∧ . . . ∧o_(n)

The above model can be formalized in HOL-Light, specifically as follows:|−MEET o ₁ o ₂ . . . o _(n) =o ₁ outer o ₂ outer . . . o _(n)

wherein o_(i) represents a geometry representing the i^(th) basiccomponent of the robot, i=1, 2, . . . , n.

The model uses an outer product to realize an intersection operation ofgeometries representing the basic components of the robot. The inputvariables of the function MEET represent any geometries on which anintersection is to be performed, and the data type is a real{circumflexover ( )}(4,1)multivector. The constraint relationship includes a jointbeing on a connecting rod (i.e., a point being on a line), a connectingrod being on an auxiliary plane on which a joint is located (i.e., aline being on a plane), and so on.

The following is a geometric pure rotation model abstracted from therotation motion of the basic components of a robot:

$\quad\left\{ \begin{matrix}{R = {{\cos\left( \frac{\phi}{2} \right)} - {L\;{\sin\left( \frac{\phi}{2} \right)}}}} \\{o_{rotated} = {{Ro}\;\overset{\sim}{R}}}\end{matrix} \right.$

The above model can be formalized in HOL-Light, specifically as follows:|−rotation_CGA t1=cos(t/&2)% mbasis{ }−1*(sin(t/&2)% mbasis{ }|−pure_rotationed_CGAx t1=(rotation_CGAt1)*x*(reversion(rotation_CGAt1))

The model uses a geometric product to realize a rotation transformationof a geometry. R is a rotation operator in conformal geometric algebra.{tilde over (R)} is a reversion of R, L represents a rotation axis, ϕ isa rotation angle, o represents the expression of the geometry beforerotation, and o_(rotated) represents the expression of the geometryafter rotation. The input variables t, l of the function rotation_CGArespectively represent a rotation angle and a rotation axis of therevolute. This function realizes the function of the rotation operator.The input variables x, t, l of the function pure_rotationed_CGArepresent the geometry, rotation angle, and rotation axis. The returnvalue of this function is a geometric representation after rotation,wherein the function reversion represents the reversion.

The following geometric pure translation model is abstracted from thetranslation motion of the basic components of a robot:

$\quad\left\{ \begin{matrix}{T = {1 = {\frac{1}{2}{te}_{\infty}}}} \\{o_{translated} = {{To}\;\overset{\sim}{T}}}\end{matrix} \right.$

The above model can be formalized in HOL-Light, specifically as follows:|−Translation_CGA t=mbasis{ }−(&1/&2)%(t*null_inf)|−pure_translationed_CGAxt=(Translation_CGA_t)*x*(reversion(Translation_CGAt))

The model uses a geometric product to realize a geometric translationtransformation. T is a translation operator in conformal geometricalgebra. {tilde over (T)} is a reversion of T, wherein t=t₁e₁+t₂e₂+t₃e₃is a translation vector, representing the direction and length of thetranslation, t₁, t₂, t₃ are coefficients, o is the expression of thegeometry before translation, o_(translated) is the expression of thegeometry after translation, and the input variable t of the functionTranslation_CGA is a translation vector, representing the direction andlength of the translation. This function realizes the function of thetranslation operator. The input variables x, t of the functionpure_translationed_CGA represent the geometry and translation vector,and the return value of the function is a geometric representation aftertranslation.

The geometric rigid body motion model is abstracted from the motion ofan end of a robot joint reaching a desired position:

$\quad\left\{ \begin{matrix}{M = {RT}} \\{o_{{rigid\_ body}{\_ motion}} = {{Mo}\;\overset{\sim}{M}}}\end{matrix} \right.$

The above model can be formalized in HOL-Light, specifically as follows:|−motor_CGA a l t=rotation_CGA a l*Translation_CGA t|−rigid_body_motion x a l t=(motor_CGA a l t)*x*(reversion(motor_CGA a lt))

This model uses a geometric product to realize the geometric rigid bodytransformation, wherein R and T are a rotation operator and atranslation operator respectively, and the two are connected by ageometric product operation. M is a Motor operator, {tilde over (M)} isa reversion of M, o is an expression of the geometry before the rigidbody motion, and o_(rigid_body_motion) represents an expression of thegeometry after the rigid body motion. The input variables x, a, l, t ofthe function motor_CGA respectively represent a geometry, a rotationangle, a rotation axis, and a translation vector. This function realizesthe function of the motor operator. The return value of the functionrigid_body_motion is a geometric representation after the rigid bodymotion. This representation indicates that the geometry is translatedfirst and then rotated, but the motor operator has associativity, andthe rotation operator and the translation operator are interchangeable.

The distance model between geometries is abstracted from the metricrelationship of the basic components of a robot:

${A \cdot B} = {{- \frac{1}{2}}{{a - b}}^{2}}$|−∀a b.(point_CGA a)inner(point_CGA b)=(−−(&1/&2)*dist(a,b)pow 2)%mbasis{ }

wherein a and b represent any two points in the three-dimensionalEuclidean space, and A and B are expressions of a point a and a point bin a conformal space respectively. The above model is an expression of apoint-to-point distance in conformal geometric algebra, whereindist(a,b) represents the distance between two points, which can berepresented by the inner product of geometric algebra. A point-to-linedistance, a point-to-sphere distance, and a sphere-to-sphere distancealso have specific expressions in conformal geometric algebra, which canbe used for specific computations in robot problems.

The geometric angle model is abstracted from an angle of the basiccomponents of a robot:

$\theta = {{\angle\left( {o_{1},o_{2}} \right)} = {{arc}\;\cos\;\frac{o_{1}^{*} \cdot o_{2}^{*}}{{o_{1}^{*}}{o_{2}^{*}}}}}$

The above model can be formalized in HOL-Light, specifically as follows:|−∀o ₁ o ₂.vector_angles_CGA o ₁ o ₂=acs((o ₁ inner o ₂)$${ }/(mult_normo ₁*mult_normo ₂))

The geometric angle model abstracted from an angle of the basiccomponents of the robot is used to compute an angle θ of the basiccomponents of the robot.

This model is an expression of the geometric angle θ in conformalgeometric algebra. The input variables o₁ and o₂ of the functionvector_angles_CGA respectively represent the geometries between which anangle is to be computed, which can be lines or planes. The data type isa real{circumflex over ( )}(4,1)multivector. The function innerrepresents the left contraction product operation in geometric algebra,$${ } represents the size of the 0-grade subspace, i.e. scalar, and thefunction mult_norm is used to perform a modulo operation formultivectors. This model can be used to compute an angle between thebasic components of the robot, with o₁*, o₂* representing duals of o₁and o₂ respectively, wherein o₁*=−o₁, o₂*=o₂.

Specifically, in an implementation of the present application,performing formal modeling for a motion process of a robot based on theestablished geometric model can include:

computing a first target circle where the end of the robot gripper islocated based on the established geometric model in a formal way;computing a second target circle where the robot gripper is locatedbased on the established geometric model and feature points on the robotgripper in a formal way; computing a translation operator of the robotgripper according to the first target circle and the second targetcircle in a formal way; computing a rotation operator of the grasping ofan object by the robot gripper according to the first target circle andthe second target circle in a formal way; computing a new targetposition of the robot gripper according to the translation operator andthe rotation operator in a formal way; realizing the formal modeling forthe motion process of the robot.

The motion process of the robot arm can include various forms. In orderto more clearly explain the embodiment of the present application, thegrasping of an object by the robot is taken as an example to describethe method for formally analyzing the motion planning of the robot armbased on the conformal geometric algebra provided by the embodiment ofthe present application, given in detail as follows.

Preferably, the specific robot in the method is an n-degree-of-freedomserial robot arm with a T-shaped robot gripper at the end of it.

The specific motion planning of a robot in the method is a motionplanning process of the grasping of an object by the robot arm.

The attribute of the motion process to be verified in the method is thatthe robot gripper successfully and firmly grasps the Object.

FIG. 2 shows a flowchart of the formal modeling of a motion planningprocess concerning the grasping of an object by a robot in the presentapplication. As shown in FIG. 2, the following steps are included.

Step 30: extracting feature points of an object and a robot gripper,respectively.

It corresponds to S30A and S30B in FIG. 2.

Step 31: computing a target circle where a grasped position of theobject is located.

It corresponds to S31 in FIG. 2.

As shown in FIGS. 3 and 4, the position and orientation of the objectare obtained by four feature points (x1, x2, x3, x4) at the edge of theobject, wherein x1, x2, and x3 are any three points on the bottom edgeof the object, and x4 is any point on the top edge of the object. Areference circle Z*_(b) formed by three points on the bottom of theobject can be obtained by a direct expression of the circle:Z* _(b) =x ₁ ∧x ₂ ∧x ₃

A plane π_(b), where the reference circle Z*_(b) on the bottom of theobject is located is:π_(b)=(Z* _(b) ∧e _(∞))I _(c)

wherein Ic is a pseudo-scalar of conformal geometric algebra, a dualoperation can be realized by the pseudo-scalar, and the dual operationcan realize a mutual transformation of two expressions of a geometry.

Since it is convenient to set the grasped position of the object in amiddle part, the target circle Z_(t) should be a circle after thereference circle on the bottom is translated by a length of

$\frac{1}{2}\left( {\pi_{b} \cdot x_{4}} \right)$in a direction −π_(b). The corresponding translation operator T wouldthen be:

$T = {1 + {\frac{1}{4}\left( {\pi_{b} \cdot x_{4}} \right)\pi_{b}e_{\infty}}}$

The target circle Z_(t) formed by the grasped part of the object wouldthen be:Z _(t) =TZ* _(b) {tilde over (T)}

{tilde over (T)} is a reversion of T.

Wherein, Z*_(b)=−Z_(b), and the above computation process is formalizedin HOL-Light as follows:

let dual_circle_zb x1 x2 x3=(circle_direct_CGA (point_CGA x1)(point_CGAx2)(point_CGA x3))

let plane_b x1 x2 x3=((dual_circle_zb x1 x2 x3) outer null_int)*pseudo

let circle_zt x1 x2 x3 x4=pure_translationed_CGA (−−DUAL (dual_circle_zbx1 x2 x3)) (−−(&1/&2)%((plane_b x1 x2 x3)inner(point_CGA x4)))

wherein the reference circle Z*_(b) at the bottom of the object callsthe direct function circle_direct_CGA of the circle, and x1, x2, x3, x4represent four feature points of the object. The HOL types are allreal{tilde over ( )}3, representing a three-dimensional Euclideanvector. Feature points are embedded from the three-dimensional Euclideanspace into the five-dimensional conformal space by the functionpoint_CGA. The target circle formed by the grasped part of the object isrealized by the pure translation function pure_translationed_CGA.

It should be noted that the target circle computed in this step can beunderstood as the first target circle above, because an end of a robotgripper generally contacts with an object when the robot gripper graspsthe object.

Step 32: computing a circle where the robot gripper is located.

It corresponds to S32 in FIG. 2.

As shown in FIG. 5, given that a center P_(h), a radius ρ of the circlewhere the robot gripper is located and two feature points a, b on therobot gripper are known, a position of the circle where the robotgripper is located can be computed. First, a position of a sphere wherethe robot gripper is located is constructed, and a standard expressionof the sphere is used to obtain the following:

$S_{h} = {P_{h} - {\frac{1}{2}\rho^{2}e_{\infty}}}$

Then, a plane π*_(h) where the robot gripper is located is computedthrough the center P_(h) and two feature points a, b on the robotgripper:π*_(h) =P _(h) ∧a∧b∧e _(∞)

The circle where the robot gripper is located is obtained by anintersection of the sphere S_(h) and the plane π*_(h) where the robotgripper is located:Z _(h) =S _(h)∧π*_(h)

The above computation process is formalized in HOL-Light as follows:

let sphere_Sh ph r=sphere_CGA ph r

let dual_plane_pih ph a b=plane_direct_CGA (point_CGA ph)(point_CGAa)(point_CGA b)

let circle_zh ph r a b=(sphere_Sh ph r)outer(−−DUAL (dual_plane_pih ph ab))

An auxiliary sphere S_(h) is realized by calling a standard expressionfunction sphere_CGA of the sphere, with (ph: real{circumflex over ( )}3)and (r: real) respectively representing the center P_(h) and the radiusr of the auxiliary sphere. An auxiliary plane π*_(h), where the robotgripper is located, is realized by calling the direct expressionfunction plane_direct_CGA of the plane, with (a: real{circumflex over( )}3) and (b: real{circumflex over ( )}3) respectively representing twopoints on the robot gripper. A dual operator uses the function DUAL.

Step 33 a: computing a translation operator of the grasping of an objectby the robot gripper, where the translation operator includes atranslation axis and a translation length.

It corresponds to S33 in FIG. 2.

It is known that the length of translation should be a distance betweenthe center of the target circle Z_(t) and the center of the circle Z_(h)where the robot gripper is located, and that the translation axis is astraight line passing through two centers. First, the center of thetarget circle Z_(t) is computed:P _(t) =Z _(t) e _(∞) Z _(t)

Since the translation axis is determined by two centers, the translationaxis l*_(T) can be computed by a direct expression:l* _(T) =P _(h) ∧P _(t) ∧e _(∞)

Accordingly, the length of translation is d:d=|l* _(T)|=dist(P _(h) ,P _(t))

The above computation process is formalized in HOL-Light:

let center_point_pt x1 x2 x3 x4=(circle_zt x1 x2 x3x4)*null_inf*(circle_zt x1 x2 x3 x4)

let dual_translation_axis ph x1 x2 x3 x4=line_direct_CGA (point_CGA ph)(center_point_pt x1 x2 x3 x4)

let distance ph x1 x2 x3 x4=−−sqrt(&2*((point_CGAph)inner(center_point_pt x1 x2 x3 x4))$${ })

Step 33 b: computing a rotation operator of the grasping of an object bythe robot gripper, with the rotation operator including a rotation axisand a rotation angle.

It corresponds to S33 in FIG. 2.

The rotation axis of the grasping of an object by the robot grippermeets a few constraints:

1. the rotation axis passes through the center P_(h) of the circle wherethe robot gripper is located;

2. the rotation axis is on a plane defined by axes of the two circles.

Therefore, two axes l*_(h) and l*_(t) of the target circle Z_(t) and thecircle Z_(h), where the robot gripper is located, are computed first:l* _(h) =Z _(h) ∧e _(∞)l* _(t) =Z _(t) ∧e _(∞)

A plane π*_(th) defined by the two axes is:π*_(th) =l* _(t)∧(l* _(h)(e ₀ ∧e _(∞)))

Then, the rotation axis is computed as follows:l* _(r) =P _(h)∧π_(th) ∧e _(∞),π_(th)=−π*_(th)

The rotation angle is known as an angle between the two axes; thus, theangle formula is used to obtain:

$\theta = {{arc}\;\cos\frac{\;{l_{t}^{*} \cdot l_{h}^{*}}}{{l_{t}^{*}}{l_{h}^{*}}}}$

The above computation process is formalized in HOL-Light as follows:

let dual_lh ph r a b=(circle_zh ph r a b) outer null_inf

let dual_lt x1 x2 x3 x4=(circle_zt x1 x2 x3 x4) outer null_inf

let dual_plane_th x1 x2 x3 x4 ph r a b=(dual_lt x1 x2 x3x4)outer((dual_lh ph r a b)*(null_zero outer null_inf))

let dual_rotation_axis x1 x2 x3 x4 ph r a b=(point_CGAph)outer(−−DUAL(dual_plane_th x1 x2 x3 x4 ph r a b))outer null_inf

let rotation angle x1 x2 x3 x4 ph r a b=vector_angles_CGA (dual_lh ph ra b) (dual_lt x1 x2 x3 x4)

Wherein the rotation angle θ is computed by using a CGA angle formula,which is realized by the function vector_angles_CGA.

Step 34: computing a new target position of the robot gripper.

It corresponds to S34 in FIG. 2.

The rotation operator R and the translation operator T of the robotgripper can be obtained by the translation axis l*_(t), the length oftranslation d, the rotation axis l*_(r), and the rotation angle θcomputed in step 33 and step 34:

${R = {{\cos\left( \frac{\theta}{2} \right)} - {l_{r}{\sin\left( \frac{\theta}{2} \right)}}}},{T = {1 - {\frac{1}{2}{dl}_{T}e_{\infty}}}}$

wherein R and T are the rotation operator and the translation operatorrespectively, and l_(r)=−l*_(r), l_(T)=−l*_(T).

The new target position of the robot gripper can be computed by rotationand then translation. Using a rigid body operator, it can be computedas:Z′ _(h) =TRZ _(h) {tilde over (R)}{tilde over (T)}

wherein Z′_(h) is the new target position of the robot gripper, {tildeover (R)} is a reversion of R, and {tilde over (T)} is a reversion of T.

Finally, the new target position of the robot gripper can be obtained,which is formalized in HOL-Light as follows:

let circle_zh_new x1 x2 x3 x4 ph r a b=pure_translationed_CGA

(pure_rotationed_CGA (circle_zh ph r a b)(rotation_angle x1 x2 x3 x4 phr a b) (−−DUAL (dual_rotation_axis x1 x2 x3 x4 ph r a b)))

((distance ph r a b x1 x2 x3 x4)%(−−DUAL(dual_translation_axis ph r a bx1 x2 x3 x4)))

Since the new target position of the robot gripper is obtained by firstrotation and then translation, a circle Z_(h) constructed by the robotgripper plane is first rotated by the pure rotation functionpure_rotationed_CGA. The input variables of this function are therotated geometry Z_(h), the rotation angle θ, and the rotation axis,i.e., (circle_zh ph r a b), (rotation_angle x1 x2 x3 x4 ph r a b), and(−−DUAL (dual_rotation axis x1 x2 x3 x4 ph r a b)). After the rotation,a translational motion is then performed for the circle RZ_(h){tildeover (R)} through the pure translation functionpure_translationed_CGAction; the input variables of this function arethe rotated geometry RZ_(h){tilde over (R)}, and a translation vectordl_(T).

The main contents of the above code involve CGA geometricrepresentation, the features of distance between the geometries, and theformalization of the geometric motion transformation. The robot grippercircle Z_(h) approaches the target circle Z_(t) step by step where aposition of the object is located. Finally, the robot grippersuccessfully and firmly grasps the object, which must meet a certaingeometric constraint relationship. In the above algorithm, theconstraint relationship is that the new position of the circle Z_(h),where the robot gripper plane is located, should coincide with theposition of the target circle Z_(t), where the object is located, thatis, equality should be met for the expressions in CGA. A goal can beestablished in HOL-Light to verify this geometric constraintrelationship:

goal: • x1 x2 x3 x4 ph r a b. circle_zh_new x1 x2 x3 x4 ph r ab=circle_zt x1 x2 x3 x4

It should be noted that the “object” mentioned in the above descriptioncan be understood as “an object that is grasped.”

In addition, after a new target position is computed in this step, acircle formed by the new target position needs to be computed andwhether the computed circle is equal to the target circle computed inthe above S31 is determined. If they are equal, this is an indicationthat the robot gripper is successfully grasping the object. Thisdetermination corresponds to S35 in FIG. 2.

The methods for performing the formal modeling and verification of themotion planning of robots, based on CGA in the solutions provided in theabove embodiments, can be loaded and used in HOL-Light. One of thebiggest difficulties encountered in the process is that the theoremproving technique requires a great deal of man-machine interaction, agreat deal of work, and a great deal of time. The process of modelingemphasizes a user's familiarity with the theoretical knowledge of CGA,and the process of verification requires rigorous thinking and someexperience with logical reasoning.

In addition, unlike the traditional method, in the technical solutionsprovided in the above embodiment, after a geometric relationship logicmodel is obtained by performing the modeling in a formal way, a motionlogic relationship corresponding to a constraint or attribute of amotion process to be verified is obtained according to the logical modelof geometric relations, and whether the above motion logic relationshipis correct is verified. The above method is accurate and complete forthe nature of the verification, since correctness is verified usingmathematical methods. In addition, CGA can be used to perform themodeling and processing on geometric elements such as points, lines,planes, circles, and spheres, and the rotation and translation of thesegeometric elements in a unified manner. It offers great advantages fordealing with problems of robot kinematics and motion planning, and canimprove dimensions for solving problems, thus simplifying the couplingin robot computations, and thus reducing the complexity of thecomputations. In view of the above two aspects, when the technicalsolutions provided by the above embodiments are applied to analyze themotion planning of a robotic arm, the complexity of the computations canbe reduced while the verification accuracy is improved. Thus, therespective advantages of CGA and of formal methods are fully exerted,and the combination of both mutually strengthens their respectiveadvantages.

Corresponding to the above method for formally analyzing the motionplanning of a robotic arm based on conformal geometric algebra, anembodiment of the present application further provides a system forformally analyzing the motion planning of a robotic arm based onconformal geometric algebra.

Specifically, the system includes:

a parameter determining module, configured for determining the specificstructural parameters and motion planning parameters of a robot;

a module for the establishing of robot basic geometric logical models,configured for establishing a corresponding geometric model for thebasic components and motion planning constraints of the robot accordingto the specific structural parameters and the motion planning parametersbased on a conformal geometric algebra theory, wherein the establishedgeometric model is described in a higher-order logic language;

a module for the establishing of a logical model of geometric relationsinvolved in the motion process of the robot, configured for performingformal modeling for a motion process of the robot based on theestablished geometric model, to obtain a logical model of geometricrelations involved in the motion process of the robot;

a module for the construction of logical propositions, configured forobtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relation;

a verification module, configured for verifying whether the motion logicrelationship is correct; if the motion logic relationship is correct,this indicates that the logical model of geometric relations meets theconstraint of the motion process to be verified or has the attribute ofthe motion process to be verified, or if the motion logic relationshipis not correct, this indicates that the logical model of geometricrelations does not meet the constraint of the motion process to beverified or does not have the attribute of the motion process to beverified.

In addition, unlike the traditional method, in the technical solutionprovided in the embodiment, after a logical model of geometric relationsis obtained by performing the modeling in a formal way, a motion logicrelationship corresponding to a constraint or attribute of a motionprocess to be verified is obtained according to the logical model ofgeometric relations, and whether the above motion logic relationship iscorrect is verified. The above method is accurate and complete for thenature of the verification, since the correctness is verified usingmathematical methods. In addition, CGA can be used to perform themodeling and processing on geometric elements such as points, lines,planes, circles, and spheres, and the rotation and translation of thesegeometric elements in a unified manner. It offers great advantages fordealing with problems of robot kinematics and motion planning, and canimprove dimensions for solving problems, thus simplifying the couplingin robot computation, and thus reducing the complexity of thecomputations. In view of the above two aspects, when the technicalsolutions provided by the above embodiments are applied to analyze themotion planning of a robotic arm, the complexity of the computations canbe reduced while the verification accuracy is improved. Thus, therespective advantages of CGA and a formal way are fully exerted, and thecombination of both even mutually strengthens their own advantages.

Correspondingly, the present application further provides a storagemedium, which is configured to store executable program code, and theexecutable program code is executed to perform the method for formallyanalyzing the motion planning of a robotic arm based on conformalgeometric algebra. The method for formally analyzing the motion planningof a robotic arm based on conformal geometric algebra in the presentapplication includes:

determining the specific structural parameters and motion planningparameters of a robot;

establishing a corresponding geometric model for the basic componentsand motion planning constraints of the robot according to the specificstructural parameters and the motion planning parameters based on aconformal geometric algebra theory, wherein the established geometricmodel is described in a higher-order logic language;

performing formal modeling for a motion process of the robot based onthe established geometric model, to obtain a logical model of thegeometric relations involved in the motion process of a robot;

obtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relations;

verifying whether the motion logic relationship is correct; if themotion logic relationship is correct, this indicates that the logicalmodel of geometric relations meets the constraint of the motion processto be verified or has the attribute of the motion process to beverified, or if the motion logic relationship is not correct, thisindicates that the logical model of geometric relations does not meetthe constraint of the motion process to be verified or does not have theattribute of the motion process to be verified.

The program code stored in the storage medium provided in the presentembodiment is performed according to the method: after obtaining alogical model of geometric relations by performing the modeling in aformal way, obtaining a motion logic relationship corresponding to aconstraint or attribute of a motion process to be verified according tothe logical model of geometric relations, and verifying whether theabove motion logic relationship is correct. The above method is accurateand complete for the nature of the verification, since the correctnessis verified using mathematical methods. In addition, CGA can be used toperform the modeling and processing on geometric elements such aspoints, lines, planes, circles, and spheres, rotation and translation ofthese geometric elements in a unified manner. It offers great advantagesin dealing with problems of robot kinematics and motion planning, andcan improve dimensions for solving problems, thus simplifying thecoupling in robot computations, and thus reducing the complexity of thecomputations. In view of the above two aspects, when the technicalsolutions provided by the above embodiments are applied to analyze themotion planning of a robotic arm, the complexity of the computations canbe reduced while the verification accuracy is improved. Thus, therespective advantages of CGA and of formal methods are fully exerted,and the combination of both mutually strengthens their own advantages.

Correspondingly, the present application further provides an applicationprogram, which is executed to perform the method for formally analyzingthe motion planning of a robotic arm based on conformal geometricalgebra. The method for formally analyzing the motion planning of arobotic arm based on conformal geometric algebra in the presentapplication includes:

determining the specific structural parameters and motion planningparameters of a robot;

establishing a corresponding geometric model for the basic componentsand motion planning constraints of the robot according to the specificstructural parameters and the motion planning parameters based on aconformal geometric algebra theory, wherein the established geometricmodel is described in a higher-order logic language;

performing formal modeling for a motion process of the robot based onthe established geometric model, to obtain a logical model of geometricrelations involved in the motion process of the robot;

obtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relations;

verifying whether the motion logic relationship is correct; if themotion logic relationship is correct, this indicates that the logicalmodel of geometric relations meets the constraint of the motion processto be verified or has the attribute of the motion process to beverified, or if the motion logic relationship is not correct, thisindicates that the logical model of geometric relations does not meetthe constraint of the motion process to be verified or does not have theattribute of the motion process to be verified.

The application program provided in the present embodiment is performedaccording to the method: after obtaining a geometric relationship logicmodel by performing the modeling in a formal way, obtaining a motionlogic relationship corresponding to a constraint or attribute of amotion process to be verified according to the geometric relationshiplogic model, and verifying whether the above motion logic relationshipis correct. The above method is accurate and complete for the nature ofthe verification, since the correctness is verified using mathematicalmethods. In addition, CGA can be used to perform the modeling andprocessing on geometric elements such as points, lines, planes, circles,and spheres, and the rotation and translation of these geometricelements in a unified manner. It offers great advantages for dealingwith problems of robot kinematics and motion planning, and can improvedimensions for solving problems, thus simplifying the coupling in robotcomputations, and thus reducing the complexity of the computations. Inview of the above two aspects, when the technical solutions provided bythe above embodiments are applied to analyze the motion planning of arobotic arm, the complexity of the computations can be reduced while theverification accuracy is improved. Thus, the respective advantages ofCGA and a formal way are fully exerted, and the combination of both evenmutually strengthens their own advantages.

Correspondingly, the present application also provides an electronicdevice, which includes:

a processor, a memory, a communication interface, and a bus,

wherein the processor, the memory, and the communication interface areconnected and communicate with each other through the bus,

the memory stores executable program code,

the processor executes a program corresponding to the executable programcode by reading the executable program code stored in the memory, toperform the method for formally analyzing the motion planning of arobotic arm based on conformal geometric algebra. The method forformally analyzing the motion planning of a robotic arm based onconformal geometric algebra in the present application includes:

determining the specific structural parameters and motion planningparameters of a robot;

establishing a corresponding geometric model for the basic componentsand motion planning constraints of the robot according to the specificstructural parameters and the motion planning parameters based on aconformal geometric algebra theory, wherein the established geometricmodel is described in a higher-order logic language;

performing formal modeling for a motion process of the robot based onthe established geometric model, to obtain a logical model of geometricrelations involved in the motion process of the robot;

obtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relations;

verifying whether the motion logic relationship is correct; if themotion logic relationship is correct, this indicates that the logicalmodel of geometric relations meets the constraint of the motion processto be verified or has the attribute of the motion process to beverified, or if the motion logic relationship is not correct, thisindicates that the logical model of geometric relations does not meetthe constraint of the motion process to be verified or does not have theattribute of the motion process to be verified.

In the electronic device provided in the embodiment, after a logicalmodel of geometric relations is obtained by performing the modeling in aformal way, a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified is obtained according tothe logical model of geometric relations, and whether the above motionlogic relationship is correct is verified. The above method is accurateand complete for the nature of the verification, since the correctnessis verified using mathematical methods. In addition, CGA can be used toperform the modeling and processing on geometric elements such aspoints, lines, planes, circles, and spheres, and the rotation andtranslation of these geometric elements in a unified manner. It offersgreat advantages in dealing with problems of robot kinematics and motionplanning, and can improve dimensions for solving problems, thussimplifying the coupling in robot computation, and thus reducing thecomputation complexity. In view of the above two aspects, when thetechnical solutions provided by the above embodiments are applied toanalyze the motion planning of a robotic arm, the complexity of thecomputations can be reduced while the verification accuracy is improved.Thus, the respective advantages of CGA, and a formal way are fullyexerted, and the combination of both even mutually strengthens their ownadvantages.

The embodiments of a system, a storage medium, an application programand an electronic device are described briefly since they aresubstantially similar to the embodiment of the method. Related contentscan refer to the part that describes the embodiment of the method.

It should be noted that the relationship terms used here, such as“first,” “second,” and the like are only used to distinguish one entityor operation from another entity or operation, but do not necessarilyrequire or imply that there is actual relationship or order betweenthese entities or operations. Moreover, the terms “include,” “comprise,”or any variants thereof are intended to cover a non-exclusive inclusion,such that processes, methods, articles, or devices, including a seriesof elements, include not only those elements that have been listed, butalso other elements that have not specifically been listed or theelements intrinsic to these processes, methods, articles, or devices.Without further limitations, elements limited by the wording“comprise(s) a/an . . . ” do not exclude additional identical elementsin the processes, methods, articles, or devices, including the listedelements.

All of the embodiments in the description are described in a correlatedmanner, and identical or similar parts in various embodiments can referto one another. In addition, the description for each embodiment focuseson the differences from other embodiments. In particular, the embodimentof the system is described briefly, since it is substantially similar tothe embodiment of the method, and the related contents can refer to thedescription of the embodiment of the method.

It can be understood by a person skilled in the art that all or a partof steps in the implementations of the above method can be accomplishedby instructing related hardware through programs, which can be stored ina computer-readable storage medium, such as in ROM/RAM, a disk, anoptical disk, and so on.

The embodiments described above are simply preferable embodiments of thepresent application, and are not intended to limit the scope ofprotection of the present application. Any modifications, alternatives,improvements, or the like within the spirit and principle of the presentapplication shall be included within the scope of protection of thepresent application.

The invention claimed is:
 1. A method for formally analyzing a motionplanning of a robotic arm based on conformal geometric algebra, which isperformed by a computing system comprising a processor, comprising:determining specific structural parameters and motion planningparameters of a robot; establishing a corresponding geometric model forbasic components and motion planning constraints of the robot accordingto the specific structural parameters and the motion planning parametersbased on a conformal geometric algebra theory, wherein the establishedgeometric model is described in a higher-order logic language;performing formal modeling for a motion process of the robot based onthe established geometric model, to obtain a logical model of geometricrelations involved in the motion process of the robot; obtaining amotion logic relationship corresponding to a constraint or attribute ofa motion process to be verified of the robot, using the logical model ofgeometric relations; verifying whether the motion logic relationship iscorrect; when the motion logic relationship is correct, this indicatesthat the logical model of geometric relations meets the constraint ofthe motion process to be verified or has an attribute of the motionprocess to be verified, or when the motion logic relationship is notcorrect, this indicates that the logical model of geometric relationsdoes not meet the constraint of the motion process to be verified ordoes not have the attribute of the motion process to be verified; andupdating the motion planning of a robotic arm when the motion logicrelationship is not correct.
 2. The method of claim 1, wherein theestablished geometric model comprises a point model abstracted from arobot joint, a sphere model abstracted from a reachable range of an endof a robot joint, a plane model abstracted from an auxiliary plane onwhich a robot joint is located, and a line model constructed byconnecting robot joint points, a geometric intersection model abstractedfrom constraint relationship of the basic components of the robot, ageometric pure rotation model abstracted from rotation motion of thebasic components of the robot, a geometric pure translation modelabstracted from translation motion of the basic components of the robot,a geometric rigid body motion model abstracted from reaching of adesired position by an end of a robot joint, a distance model betweengeometries abstracted from the metric relationship of the basiccomponents of the robot, and a geometric angle model abstracted from anangle of the basic components of the robot.
 3. The method of claim 2,wherein the point model abstracted from a robot joint is represented asfollows: $S = {s + {\frac{1}{2}s^{2}e_{\infty}} + e_{0}}$$e_{0} = {\frac{1}{2}\left( {e_{-} - e_{+}} \right)}$ e_(∞) = e⁻ + e₊wherein s=s₁e₁+s₂e₂+s₃e₃ represents a point in a three-dimensionalEuclidean space, e₁, e₂ and e₃ are unit orthonormal bases in thethree-dimensional Euclidean space, s₁, s₂, and s₃ are coefficients, S isan expression of a point s mapped from the Euclidean space into aconformal space, e₀ represents an origin point, e_(∞) represents a pointat infinity, and e₊, e⁻ are respectively the fourth base vector and thefifth base vector in the conformal geometric space.
 4. The method ofclaim 2, wherein the sphere model abstracted from a reachable range ofan end of a robot joint is used to abstract the reachable range of theend of the robot joint into a sphere S, whose center is the joint andwhose radius is a connecting rod between the joint and another joint,which is represented as: $S = {P - {\frac{1}{2}r^{2}e_{\infty}}}$wherein P and r respectively represent the center and the radius, ande_(∞) represents a point at infinity.
 5. The method of claim 2, whereinthe plane model abstracted from an auxiliary plane on which a robotjoint is located is used to abstract the auxiliary plane on which therobot joint is located into a plane π, which is represented as:π=n+de _(∞) wherein n and d respectively represent a normal vector ofthe auxiliary plane and the distance from the auxiliary plane to theorigin point, and e_(∞) represents a point at infinity.
 6. The method ofclaim 2, wherein the line model constructed by connecting robot jointpoints is used to abstract a connecting rod between two joints into aline L*, which is represented as:L*=A∧B∧e _(∞) wherein A and B respectively represent points representedby the two joints, and e_(∞) represents a point at infinity.
 7. Themethod of claim 2, wherein the geometric intersection model o abstractedfrom the constraint relationship of the basic components of the robot isrepresented as:o=o ₁ ∧o ₂ ∧ . . . o _(n) wherein of represents a geometry representingthe ith basic component of the robot, i=1, 2, . . . , n.
 8. The methodof claim 2, wherein the geometric pure rotation model abstracted fromthe rotation motion of the basic components of the robot is representedas: $\quad\left\{ \begin{matrix}{R = {{\cos\left( \frac{\phi}{2} \right)} - {L\;{\sin\left( \frac{\phi}{2} \right)}}}} \\{o_{rotated} = {{Ro}\;\overset{\sim}{R}}}\end{matrix} \right.$ wherein R is a rotation operator in the conformalgeometric algebra, {tilde over (R)} is a reversion of R, L is a rotationaxis, ϕ is a rotation angle, o is an expression of the geometry beforerotation, and o_(rotated) is an expression of the geometry afterrotation.
 9. The method of claim 2, wherein the geometric puretranslation model abstracted from the translation motion of the basiccomponents of the robot is represented as: $\quad\left\{ \begin{matrix}{T = {1 - {\frac{1}{2}{te}_{\infty}}}} \\{o_{translated} = {{To}\;\overset{\sim}{T}}}\end{matrix} \right.$ wherein T is a translation operator in theconformal geometric algebra and T is a reversion of T, whereint=t₁e₁+t₂e₂+t₃e₃ is a translation vector representing the direction andlength of the translation, o is an expression of the geometry beforetranslation, o_(translated) is an expression of the geometry aftertranslation, and eco represents a point at infinity.
 10. The method ofclaim 2, wherein the geometric rigid body motion model abstracted fromthe reaching of a desired position by an end of a robot joint isrepresented as: $\quad\left\{ \begin{matrix}{M = {RT}} \\{o_{{rigid\_ body}{\_ motion}} = {{Mo}\;\overset{\sim}{M}}}\end{matrix} \right.$ wherein R and T are a rotation operator and atranslation operator respectively, M is a Motor operator, {tilde over(M)} is a reversion of M, o is an expression of the geometry beforerigid body motion, and o_(rigid_body_motion) is an expression of thegeometry after rigid body motion.
 11. The method of claim 2, wherein thedistance model between geometries abstracted from the metricrelationship of the basic components of the robot is represented as:${A \cdot B} = {{- \frac{1}{2}}{{a - b}}^{2}}$ wherein a, b representany two points in the three-dimensional Euclidean space, and A, B areexpressions of points a, b in a conformal space respectively.
 12. Themethod of claim 2, wherein the geometric angle model abstracted from anangle of the basic components of the robot is used to compute an angle θof the basic components of the robot, which is represented as:$\theta = {{\angle\left( {o_{1},o_{2}} \right)} = {{arc}\;\cos\;\frac{o_{1}^{*} \cdot o_{2}^{*}}{{o_{1}^{*}}{o_{2}^{*}}}}}$wherein o₁, o₂ respectively represent geometries between which an angleis to be computed, and o₁*, o₂* respectively represent duals of o₁ ando₂, wherein o₁*=−o₁, o₂*=o₂.
 13. The method of claim 1, whereinperforming formal modeling for a motion process of the robot based onthe established geometric model comprises: computing a first targetcircle where an end of a robot gripper is located based on theestablished geometric model in a formal way; computing a second targetcircle where the robot gripper is located based on the establishedgeometric model and feature points on the robot gripper in a formal way;computing a translation operator of the robot gripper according to thefirst target circle and the second target circle in a formal way;computing a rotation operator of grasping of an object by the robotgripper according to the first target circle and the second targetcircle in a formal way; computing a new target position of the robotgripper according to the translation operator and the rotation operatorin a formal way, to realize the formal modeling of the motion process ofthe robot.
 14. The method of claim 13, the motion process of the robot,comprises: a motion process of the grasping of an object by the robot;wherein the first target circle is computed as follows: a position andorientation of the object that is grasped are obtained by four featurepoints x1, x2, x3, and x4 at an edge of the object that is grasped,wherein x1, x2, and x3 are any three points on a bottom edge of theobject that is grasped and x4 is any point on a top edge of the objectthat is grasped; and a reference circle Z_(b)* formed by three points onthe bottom of the object that is grasped is obtained by the followingdirect expression of the circle:Z _(b) *=x ₁ ∧x ₂ ∧x ₃; a plane π_(b), where the reference circle Z_(b)*on the bottom of the object that is grasped is located, is obtained asfollows:π_(b)=(Z _(b) *∧e _(∞))I _(c); wherein I_(c) is a pseudo-scalar of theconformal geometric algebra, and e_(∞) represents a point at infinity; athird target circle Z_(t), where a grasped position of the object thatis grasped is located, is a circle after the reference circle istranslated by a length of$\frac{1}{2}\left( {\pi_{b} \cdot x_{4}} \right)$ in a direction of−π_(b); then, the corresponding translation operator T is:${T = {1 + {\frac{1}{4}\left( {\pi_{b} \cdot x_{4}} \right)\pi_{b}e_{\infty}}}};$the third target circle Z_(t) computed in a formal way is:Z _(t) =TZ _(b) *{tilde over (T)}; wherein the {tilde over (T)} is areversion of T; the third target circle is determined as the firsttarget circle.
 15. The method of claim 14, wherein the second targetcircle where the robot gripper is located is computed as follows: giventhat a center P_(h), a radius p of the circle where the robot gripper islocated, and two feature points a, b on the robot gripper are known, aposition of the circle where the robot gripper is located is computed;first, a position of a sphere where the robot gripper is located isconstructed, and a standard expression of the sphere is used to obtain:${S_{h} = {P_{h} - {\frac{1}{2}\rho^{2}e_{\infty}}}};$ wherein S_(h) isthe sphere where the robot gripper is located; a plane π_(h)* where therobot gripper is located is computed according to the two feature pointsa, b on the robot gripper:π_(h) *=P _(h) ∧a∧b∧e _(∞); an intersection of the sphere S_(h) and theplane π_(h)* where the robot gripper is located is computed in a formalway to obtain a second target circle Z_(h), where the robot gripper islocated:Z _(h) =S _(h)∧π_(h)*.
 16. The method of claim 15, wherein thetranslation operator of the robot gripper comprises a translation axisand a translation length, which are specifically computed as follows:first, compute the center of the first target circle Z_(t):P _(t) =Z _(t) e _(∞) Z _(t); compute the translation axis l_(T)* in aformal way and a direct expression:l _(T) *=P _(h) ∧P _(t) ∧e _(∞); compute a length of translation d in aformal way:d=|l _(T)*|=dist(P _(h) ,P _(t)).
 17. The method of claim 16, whereinthe rotation operator of the robot gripper comprises a rotation axis anda rotation angle, which are specifically computed as follows: computetwo axes l_(h)* and l_(t)* of the first target circle Z_(t) and thesecond target circle Z_(h):l _(h) *=Z _(h) ∧e _(∞) ,l _(t) *=Z _(t) ∧e _(∞); the plane π_(th)*defined by the two axes is:π_(th) *=l _(t)*∧(l _(h)*(e ₀ ∧e _(∞))); wherein e₀ represents an originpoint, and e_(∞) represents a point at infinity; compute the rotationaxis l_(r)* in a formal way: l_(r)*=P_(h)∧π_(th)∧e_(∞), π_(th)=−π_(th)*;compute the rotation angle in a formal way:$\theta = {{arc}\;\cos\;{\frac{l_{t}^{*} \cdot l_{h}^{*}}{{l_{t}^{*}}{l_{h}^{*}}}.}}$18. The method of claim 17, wherein the new target position of the robotgripper is computed as follows: the rotation operator and thetranslation operator of the grasping of the object by the robot gripperare obtained by the translation axis l_(T)*, the length of translation,the rotation axis l_(r)*, and the rotation angle θ:${R = {{\cos\left( \frac{\theta}{2} \right)} - {l_{r}{\sin\left( \frac{\theta}{2} \right)}}}},{{T = {1 - {\frac{1}{2}{dl}_{T}e_{\infty}}}};}$wherein R and T are the rotation operator and the translation operatorrespectively, and l_(r)=−l_(r)*, l_(T)=−l_(T)*; through a formal way,the new target position of the robot gripper is computed as follows:Z _(h) ′=TRZ _(h) {tilde over (R)}{tilde over (T)}; wherein Z_(h)′ isthe new target position of the robot gripper, {tilde over (R)} is areversion of R, and {tilde over (T)} is a reversion of T.
 19. A systemfor formally analyzing a motion planning of a robotic arm based onconformal geometric algebra, the system comprising a computing systemcomprising a processor configured to: determine specific structuralparameters and motion planning parameters of a robot; establish acorresponding geometric model for basic components and motion planningconstraints of the robot according to the specific structural parametersand the motion planning parameters based on a conformal geometricalgebra theory, wherein the established geometric model is described ina higher-order logic language; establish a logical model of geometricrelations involved in the motion process of the robot, configured forperforming formal modeling for a motion process of the robot based onthe established geometric model, to obtain a logical model of geometricrelations involved in the motion process of the robot; obtain a motionlogic relationship corresponding to a constraint or attribute of amotion process to be verified of the robot, using the logical model ofgeometric relations; verify whether the motion logic relationship iscorrect; when the motion logic relationship is correct, this indicatesthat the logical model of geometric relations meets the constraint ofthe motion process to be verified or has an attribute of the motionprocess to be verified, or when the motion logic relationship is notcorrect, this indicates that the logical model of geometric relationsdoes not meet the constraint of the motion process to be verified ordoes not have the attribute of the motion process to be verified; andupdate the motion planning of a robotic arm when the motion logicrelationship is not correct.
 20. An electronic device comprising: aprocessor, a memory, a communication interface, and a bus, wherein theprocessor, the memory, and the communication interface are connected andcommunicate with each other through the bus, the memory stores anexecutable program code, the processor executes a program correspondingto the executable program code by reading the executable program codestored in the memory, to perform the method for formally analyzing themotion planning of a robotic arm based on conformal geometric algebra,comprising: determining specific structural parameters and motionplanning parameters of a robot; establishing a corresponding geometricmodel for basic components and motion planning constraints of the robotaccording to the specific structural parameters and the motion planningparameters based on a conformal geometric algebra theory, wherein theestablished geometric model is described in a higher-order logiclanguage; performing formal modeling for a motion process of the robotbased on the established geometric model, to obtain a logical model ofgeometric relations involved in the motion process of the robot;obtaining a motion logic relationship corresponding to a constraint orattribute of a motion process to be verified of the robot, using thelogical model of geometric relations; verifying whether the motion logicrelationship is correct; when the motion logic relationship is correct,this indicates that the logical model of geometric relations meets theconstraint of the motion process to be verified or has an attribute ofthe motion process to be verified, or when the motion logic relationshipis not correct, this indicates that the logical model of geometricrelations does not meet the constraint of the motion process to beverified or does not have the attribute of the motion process to beverified; and updating the motion planning of a robotic arm when themotion logic relationship is not correct.